Field-induced quantum criticality in low-dimensional Heisenberg spin systems Mohamed Azzouz* Department of Physics and Astronomy, Laurentian University, Ramsey Lake Road, Sudbury, Ontario, Canada P3E 2C6 Received 24 May 2006; revised manuscript received 12 September 2006; published 20 November 2006 We study the quantum critical behavior in the antiferromagnetic Heisenberg chain and two-leg Heisenberg ladder resulting from the application of an external magnetic field. In each of these systems a finite-temperature crossover line between two different ferromagnetic phases ends with a quantum critical point at zero tempera- ture. Using the bond-mean-field theory, we calculate the field dependence of the magnetization and the mean- field spin bond parameters in both systems. For the Heisenberg chain, we recover the existing exact results and show in addition that the saturation of the zero-temperature magnetization at the field h c =2J is accompanied by a quantum phase transition, where the bond parameter vanishes. Here J is the exchange coupling constant along the chain. For the two-leg ladder, we also recover the known results, like the two magnetization plateaus, and show that at the upper critical field, which corresponds to the appearance of the saturation magnetization plateau, the chain and rung spin bond parameters vanish. The identification of the order parameters that govern the field-induced quantum criticality in the systems we study here constitutes an original contribution. Because no long-range order, which breaks symmetry, characterizes the bond order, the latter could be a proposal for the so-called hidden order. We calculate analytically the bond parameters in both systems as functions of the field in the low- and high-field limits at zero temperature. At nonzero temperatures, the calculation of the magne- tization and bond parameters is carried out by solving the mean-field equations numerically. DOI: 10.1103/PhysRevB.74.174422 PACS numbers: 75.10.Jm, 75.30.m, 73.43.Nq I. INTRODUCTION According to Sachdev, a second-order quantum phase transition occurs in a system at absolute zero temperature when an order parameter vanishes at a critical value of a an external coupling parameter, like pressure or magnetic field. 1 Unlike classical phase transitions, which are driven by thermal fluctuations, a quantum phase transition is driven solely by quantum fluctuations. In this paper, we examine the occurrence of quantum criticality in two low-dimensional an- tiferromagnetic AF Heisenberg spin systems, which is in- duced by an external magnetic field. These systems are the Heisenberg chain and the two-leg Heisenberg ladder. To the best of our knowledge, quantum criticality was not addressed in these systems in the way we do here. We identify the microscopic parameters that govern the zero-temperature phase transition in these systems. These parameters are found to vary with, and vanish at, a critical value of the magnetic field. Bonner and Fisher 2 and Parkinson and Bonner 3 have cal- culated the zero-temperature magnetization versus the mag- netic field for the one-dimensional 1D Heisenberg model and found that the magnetization reaches the saturation value M s at the critical field h c =2J according to the law M z M s =1- 4   1- h h c  1/2 , 1 which is valid for fields h slightly smaller than h c . J is the spin exchange coupling constant. Note that for the Heisenberg chain the magnetization does not vanish abruptly at any value of the field; it increases linearly with the field near zero, then saturates at h c . By analyzing the spin configu- ration, one finds that the latter evolves from the well- understood ground state with short-range AF order in zero field a gapless spin liquid with algebraically decaying AF correlations and a correlation length  =  to the ferromag- netic state with magnetization linearly increasing with field near zero, then finally to the fully saturated ferromagnetic ground state when h  h c . 2,3 We find that the competition between the Ising and Zeeman interactions plays an impor- tant role in the way the magnetization depends on the field and that the unsaturated ferromagnetic state for 0  h  h c is governed not only by the magnetization, but also by a pa- rameter reflecting the AF Ising interaction and the quantum fluctuations: i.e., the spin bond parameter. This bond param- eter decreases with increasing field, then vanishes at the criti- cal value h c , at which ferromagnetism reaches saturation. The unsaturated ferromagnetic state for 0  h  h c can be called a quantum ferromagnet because quantum fluctuations prevent the saturation of the magnetic moment. For h  h c the ground state can be said to have classical ferromagnetism because the quantum fluctuations and Ising interaction are both irrelevant in this state. In sum, we show in this work that the saturation of the magnetization in the Heisenberg chain is accompanied by a second-order phase transition and identify and calculate the field dependence of the order pa- rameter governing this phase transition. The two-leg Heisenberg ladder is characterized by two bond parameters, one along the chains and the second one along the rungs. Also in this case, these bond parameters vanish when ferromagnetism reaches saturation at the upper critical field h c2 , which depends on the rung coupling. It is known that the two-leg Heisenberg ladder presents two mag- netization plateaus: one zero-magnetization plateau for fields smaller than the lower critical field h c1 = E g , where E g is the energy gap, and the other plateau in the saturated ferromag- netic regime realized for fields greater than h c2  h c1 . 4 Chitra and Giamarchi, 5 among many others, studied the two-leg ladder using the bosonization technique. They focused on the fields near h c1 and argued that a quantum critical transition occurs at this field. While we do not disagree with Chitra and PHYSICAL REVIEW B 74, 174422 2006 1098-0121/2006/7417/17442212 ©2006 The American Physical Society 174422-1